## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 4 (2017), 2130-2158.

### Metastability for Glauber dynamics on random graphs

S. Dommers, F. den Hollander, O. Jovanovski, and F. R. Nardi

#### Abstract

In this paper, we study metastable behaviour at low temperature of Glauber spin–flip dynamics on random graphs. We fix a large number of vertices and randomly allocate edges according to the configuration model with a prescribed degree distribution. Each vertex carries a spin that can point either up or down. Each spin interacts with a positive magnetic field, while spins at vertices that are connected by edges also interact with each other via a ferromagnetic pair potential. We start from the configuration where all spins point down, and allow spins to flip up or down according to a Metropolis dynamics at positive temperature. We are interested in the time it takes the system to reach the configuration where all spins point up. In order to achieve this transition, the system needs to create a sufficiently large droplet of up-spins, called critical droplet, which triggers the crossover.

In the limit as the temperature tends to zero, and subject to a certain *key hypothesis* implying metastable behaviour, the average crossover time follows the classical *Arrhenius law*, with an exponent and a prefactor that are controlled by the *energy* and the *entropy* of the critical droplet. The crossover time divided by its average is exponentially distributed. We study the scaling behaviour of the exponent as the number of vertices tends to infinity, deriving upper and lower bounds. We also identify a regime for the magnetic field and the pair potential in which the key hypothesis is satisfied. The critical droplets, representing the saddle points for the crossover, have a size that is of the order of the number of vertices. This is because the random graphs generated by the configuration model are expander graphs.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 4 (2017), 2130-2158.

**Dates**

Received: February 2016

Revised: August 2016

First available in Project Euclid: 30 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1504080028

**Digital Object Identifier**

doi:10.1214/16-AAP1251

**Mathematical Reviews number (MathSciNet)**

MR3693522

**Zentralblatt MATH identifier**

1377.60020

**Subjects**

Primary: 60C05: Combinatorial probability 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 82C27: Dynamic critical phenomena

**Keywords**

Random graph Glauber spin–flip dynamics metastability critical droplet configuration model

#### Citation

Dommers, S.; den Hollander, F.; Jovanovski, O.; Nardi, F. R. Metastability for Glauber dynamics on random graphs. Ann. Appl. Probab. 27 (2017), no. 4, 2130--2158. doi:10.1214/16-AAP1251. https://projecteuclid.org/euclid.aoap/1504080028